After showing the downwards density of nonhemimaximal degrees, Downey and Stob continued to prove that the existence of a low₂, but not low, nonhemimaximal degree, and their proof uses the fact that incomplete m-topped degrees are low₂ but not low. As commented in their paper, the construction of such a nonhemimaximal degree is actually a primitive 0''' argument. In this paper, we give another construction of such degrees, which is a standard 0''-argument, much simpler than Downey and Stob's construction mentioned above.
"Nonhemimaximal degrees and the high/low hierarchy." J. Symbolic Logic 77 (2) 433 - 446, June 2012. https://doi.org/10.2178/jsl/1333566631